Room P3.10, Mathematics Building

Sean Lawton, Instituto Superior Técnico
Obtaining the One-Holed Torus from Pants: Duality in an $SL(3,\mathbb{C})$-Character Variety

The moduli of representations from a surface group into a reductive algebraic group $G$ is a poisson variety. However if two surfaces with boundary have the same Euler characteristic then their moduli are algebraically identical. They are distinguished by their symplectic leaves. When $G=SL(3,\mathbb{C})$, we show that the symplectic leaves for the one-holed torus are generically transverse to the symplectic leaves of the three-holed sphere. The projection mapping the three-holed sphere to the one-holed torus induces a mapping of the coordinate rings of the moduli which kills the poisson bivector of the one-holed torus. Consequently, since the tangent space of a representation breaks up into a direct sum of two symplectic vector spaces, the induced bivector projects to the bivector of the three-holed sphere under projection. This talk reflects work in progress.